# How many noncyclic submodules with $9$ elements does $V$ have?

Let $$V=\mathbb{Z}/(3) \oplus \mathbb{Z}/(3) \oplus \mathbb{Z}/(9)$$.

1. How many submodules with $$3$$ elements does $$V$$ have?

Because $$\phi(3)=2$$, each subgroup of order $$3$$ has $$2$$ elements of order $$3$$. Since no $$2$$ cyclic subgroup can have an element of order $$3$$ in common, there $$6\div 2=3$$ cyclic subgroups. Note that there are no non-cyclic subgroups with order $$3$$ because $$\mathbb{Z}_3$$ is always cyclic.

1. How many of the submodules $$W$$ of $$V$$ with $$3$$ elements have a complementary direct summand, i.e., are such that there exists a submodule $$W'$$ of $$V$$ with $$V=W\oplus W'$$?

I want to say that $$W'=\mathbb{Z}_3\oplus \mathbb{Z}_9$$ but it feels like I am guessing. Also how can I determine how many submodules there are with this property?

1. How many cyclic submodules with $$9$$ elements does $$V$$ have?

Because $$\phi(9)=6$$, each subgroup of order $$9$$ has $$6$$ elements of order $$9$$. Since no $$2$$ cyclic subgroups can have an element of order $$9$$ in common, there are $$54\div 6=9$$ cyclic subgroups.

1. How many noncyclic submodules with $$9$$ elements does $$V$$ have?

The noncyclic subgroups with order $$9$$ are isomorphic to $$\mathbb{Z}_3\oplus \mathbb{Z}_3$$. i.e. In a non-cyclic subgroups of order $$3$$ each non-identity element is of order $$3$$.

Where do I go from here?